Not necessarily, I did that as a kid in the late 2000s. My friend's parents had an old mac in their basement that we would play flash games and watch stupid youtube videos on.
83 baby here. Perfect timing. Grew up during the early internet, before Facebook and phone cameras. No such thing as online bullying and nobody could film you getting beaten up.
Born in 2000, my parents had a computer (running Windows XP) but it was only for work. Went over to my friends' houses to experience the information superhighway.
I did it being born in 94. It wasn't about who has access to the internet, it's that I wanted to hang out with my friend in person like a normal 10 year old but the Internet was the coolest thing to do at the time.
Hell, friends and I were doing it 2008 in college. 6 or 7 of us all gathered around a single 24" monitor watching the latest episode from Nostalgia Critic or something similar.